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Quaternion Algebraic Geometry

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Quaternion Algebraic Geometry Quaternion Algebraic Geometry Dominic Widdows June 5, 2006 Abstract QUATERNION ALGEBRAIC GEOMETRY DOMINIC WIDDOWS St Anne’s College, Oxford Thesis submitted Hilary Term, 2000, in support of application to supplicate for the degree of D.Phil. This thesis is a collection of results about hypercomplex and quaternionic manifolds, focussing on two main areas. These are exterior forms and double complexes, and the ‘algebraic geometry’ of hypercomplex manifolds. The latter area is strongly influenced by techniques from quaternionic algebra. A new double complex on quaternionic manifolds is presented, a quaternionic version of the Dolbeault complex on a complex manifold. It arises from the decomposition of real-valued exterior forms on a quaternionic manifold M into irreducible representations of Sp(1). This decomposition gives a double complex of differential forms and operators ∼ as a result of the Clebsch-Gordon formula Vr⊗V1 = Vr+1⊕Vr−1 for Sp(1)-representations. The properties of the double complex are investigated, and it is established that it is elliptic in most places. Joyce has created a new theory of quaternionic algebra [J1] by defining a quaternionic tensor product for AH-modules (H-modules equipped with a special real subspace). The 1 theory can be described using sheaves over CP , an interpretation due to Quillen [Q]. AH-modules and their quaternionic tensor products are classified. Stable AH-modules are described using Sp(1)-representations. This theory is especially useful for describing hypercomplex manifolds and forming close analogies with complex geometry. Joyce has defined and investigated q-holomorphic functions on hypercomplex manifolds. There is also a q-holomorphic cotangent space which again arises as a result of the Clebsch-Gordon formula. AH-module bundles are defined and their q-holomorphic sections explored. Quaternion-valued differential forms on hypercomplex manifolds are of special inter- est. Their decomposition is finer than that of real forms, giving a second double complex with special advantages. The cohomology of these complexes leads to new invariants of compact quaternionic and hypercomplex manifolds. Quaternion-valued vector fields are also studied, and lead to the definition of quater- nionic Lie algebras. The investigation of finite-dimensional quaternionic Lie algebras allows the calculation of some simple quaternionic cohomology groups. Acknowledgements I would like to express gratitude, appreciation and respect for my supervisor Do- minic Joyce. Without his inspiration as a mathematician this thesis would not have been conceived; without his patience and friendship it would certainly not have been completed. Dedication To the memory of Dr Peter Rowe, 1938-1998, who taught me relativity as an undergraduate in Durham. His determination to teach concepts before examination techniques introduced me to differential geometry. i Contents Introduction 1 1 The Quaternions and the Group Sp(1) 4 1.1 The Quaternions . 4 1.2 The Lie Group Sp(1) and its Representations . 9 1.3 Difficulties with the Quaternions . 12 2 Quaternionic Differential Geometry 17 2.1 Complex, Hypercomplex and Quaternionic Manifolds . 17 2.2 Differential Forms on Complex Manifolds . 20 2.3 Differential Forms on Quaternionic Manifolds . 23 3 A Double Complex on Quaternionic Manifolds 25 3.1 Real forms on Complex Manifolds . 25 3.2 Construction of the Double Complex . 28 3.3 Ellipticity and the Double Complex . 32 3.4 Quaternion-valued forms on Hypercomplex Manifolds . 40 4 Developments in Quaternionic Algebra 42 4.1 The Quaternionic Algebra of Joyce . 42 4.2 Duality in Quaternionic Algebra . 48 4.3 Real Subspaces of Complex Vector Spaces . 52 4.4 K-modules . 53 4.5 The Sheaf-Theoretic approach of Quillen . 56 5 Quaternionic Algebra and Sp(1)-representations 61 5.1 Stable AH-modules and Sp(1)-representations . 61 5.2 Sp(1)-Representations and the Quaternionic Tensor Product . 68 5.3 Semistable AH-modules and Sp(1)-representations . 74 5.4 Examples and Summary of AH-modules . 77 6 Hypercomplex Manifolds 80 6.1 Q-holomorphic Functions and H-algebras . 81 6.2 The Quaternionic Cotangent Space . 86 6.3 AH-bundles . 90 6.4 Q-holomorphic AH-bundles . 93 ii 7 Quaternion Valued Forms and Vector Fields 100 7.1 The Quaternion-valued Double Complex . 101 7.2 Vector Fields and Quaternionic Lie Algebras . 109 7.3 Hypercomplex Lie groups . 112 References 117 iii Introduction This aim of this thesis is to describe and develop various aspects of quaternionic algebra and geometry. The approach is based upon two pillars, namely the differential geometry of quaternionic manifolds and Joyce’s recent theory of quaternionic algebra. Contribu- tions are made to both fields of study, enabling these strands to be woven together in describing the algebraic geometry of hypercomplex manifolds. A recurrent theme throughout will be representations of the group Sp(1) of unit quaternions. Our contribution to the theory of quaternionic manifolds relies on decom- posing the Sp(1)-action on exterior forms, whilst the main new insight in quaternionic algebra is that the most important building blocks of Joyce’s theory are best described and manipulated as Sp(1)-representations. The importance of Sp(1)-representations to both areas is chiefly responsible for the successful synthesis of methods in the work on hypercomplex manifolds. Another frequent source of motivation is the behaviour of the complex numbers. Many situations in complex algebra and geometry have interesting quaternionic ana- logues. Aspects of complex geometry can often be described using the group U(1) of unit complex numbers; replacing this with the group Sp(1) can lead directly to quaternionic versions. The decomposition of exterior forms on quaternionic manifolds is precisely such an example, as is all the work on q-holomorphic functions and forms on hypercom- plex manifolds. On the other hand, Joyce’s quaternionic algebra is such a rich theory precisely because real subspaces of quaternionic vector spaces behave so differently from real subspaces of complex vector spaces. Much of the original work presented is enticingly simple — indeed, I have often felt both surprised and privileged that it has not been carried out before. One of the main explanations for this is the relative unpopularity suffered by the quaternions in the 20th century. This situation has left various aspects of quaternionic behaviour unexplored. To help understand the reasons for this omission and the consequent opportunities for development, the first chapter is devoted to a survey of the history of the quaternions and their applications. Background material also includes an introduction to the group Sp(1) and its representations. The irreducible representations on complex vector spaces and their tensor products are described, as are real and quaternionic representations. Chapter 2 is about quaternionic structures in differential geometry. The approach is based on the work of Salamon [S3]. Taking complex manifolds as a model, hyper- complex manifolds (those possessing a torsion-free GL(n, H)-structure) are defined, fol- lowed by the broader class of quaternionic manifolds (those possessing a torsion-free Sp(1)GL(n, H)-structure). After reviewing the Dolbeault complex, we consider the decomposition of differential forms on quaternionic manifolds, including an important elliptic complex discovered by Salamon upon which the integrability of the quaternionic 1 structure depends. This complex is in fact the top row of a hitherto undiscovered double complex on quaternionic manifolds, which is the subject of Chapter 3. As an Sp(1)-representation, 4n ∗ ∼ the cotangent space of a quaternionic manifold M takes the form T M = 2nV1, where 2 V1 is the basic representation of Sp(1) on C . Decomposition of the induced Sp(1)- representation on ΛkT ∗M is a simple process achieved by considering weights. That this decomposition gives rise to a double complex results from the Clebsch-Gordon formula ∗ ∼ ∼ Vr ⊗T M = Vr ⊗2nV1 = 2n(Vr+1 ⊕Vr−1). The new double complex is shown to be elliptic everywhere except along its bottom row, consisting of the basic representations V1 and the trivial representations V0. This double complex presents us with new quaternionic cohomology groups. In the fourth chapter (which is partly a summary of the work of Joyce [J1] and Quillen [Q]) we move to our other major area of interest, the theory of quaternionic algebra. The building blocks of this theory are H-modules equipped with a special real subspace. Such an object is called an AH-module. Joyce has discovered a canonical tensor product operation for AH-modules which is both associative and commutative. Using ideas from Quillen’s work, we classify AH-modules up to isomorphism. Dual AH-modules are defined and shown to have interesting properties. Particularly well- behaved is the category of stable AH-modules. In Chapter 5 it is shown that all stable AH-modules and their duals are conveniently described using Sp(1)-representations. The resulting theory is ideally adapted for describing hypercomplex geometry, a pro- cess begun in Chapter 6. A hypercomplex manifold M has a triple of global anticom- muting complex structures which can be identified with the imaginary quaternions. This identification enables tensors on hypercomplex manifolds to be treated using the tech- niques of quaternionic algebra. Joyce has already used such an approach to define and investigate q-holomorphic functions on hypercomplex manifolds, which are seen as the quaternionic analogue of holomorphic functions. There is a natural product map on the AH-module of q-holomorphic functions, which gives the q-holomorphic functions an algebraic structure which Joyce calls an H-algebra. Using the Sp(1)-version of quaternionic algebra, we define a natural splitting of the ∗ ∼ quaternionic cotangent space H ⊗ T M = A ⊕ B, and show that q-holomorphic func- tions are precisely those whose differentials take values in A ⊂ H ⊗ T ∗M. The bundle A is hence defined to be the q-holomorphic cotangent space of M.
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